1. CSE543T: Algorithms for Nonlinear Optimization Yixin Chen Department of Computer Science & Engineering Washington University in St Louis Spring, 2011

2. Nonlinear Programming (NLP) problem Minimize f(x) (optional) Subject to h(x) =0 g(x) <= 0 Characteristics: –variable space X: continuous, discrete, mixed, curves Decision variables, control variables, system inputs –function properties Closed form, evaluation process, stochastic, …

3. Why non-linear? Deal or no deal? You can choose –Take $2M and go home; Or, –Deal: could get $0.1 or $5M –What will you choose?

4. Why study nonlinear optimization? Optimization is everywhere –Constraints are everywhere –The world is nonlinear (curvature and local optima) Plenty of applications –Investment, networking design, machine learning (NN, HMM, CRF), data mining, sensors, structural design, bioinformatics, medical treatment planning, games Nonlinear optimization is difficult –Curse of dimensionality, among many other reasons

5. Example: Linear Regression

6. Logistic Regression Y=0 or 1 Where nonlinear functions clearly make more sense P(Y=1) = 1/(1+exp(-w T x)) Find w to maximize ∏ i=1..n P(Y i =1) Yi (1-P(Y i =1)) (1-Yi)

8. SVM Minimize || w|| subject to: y i (w x i - b) ≥ 1

9. Sensor network optimization Variables –Number of sensors –Locations of sensors Functions –Minimize the maximum false alarming rate –Subject to minimum detection probability > 95% –Highly nonlinear functions Based on a Gaussian model and voting procedure Computed by a Monte-Carlo simulation

10. Sodoku

11. Goals of the course Introduce the theory and algorithms for nonlinear optimization –One step further to the back of the blackbox –Able to hack solvers –But not too much gory details of mathematics Hands-on experiences with optimization packages –Know how to choose solvers –Understanding of the characteristics of problems/solvers Mathematical modeling –Modeling languages –CSE applications Help solve problems in your own research

12. Overview of the course Unconstrained optimization (6 lectures) Convex constrained optimization (3 lectures) General constrained optimization (12-14 lectures) –Penalty methods –Lagrange methods –Dual methods

13. Assignments and grading Three or four homework (40%) One project (30%) –In-class presentations –Reports One final exam (30%)